From a Table to an Approximation
A table gives selected values of a function, not a picture of everything the function does between those values. Even so, when a partition and the needed sample values are provided, the table can supply the rectangle heights or trapezoid endpoint heights for a numerical approximation of an integral.
The main task is to match each value to its role. A left sum uses the function values at the left ends of the subintervals. A right sum uses the right ends. A midpoint sum uses the centers. A trapezoidal sum uses both endpoints of every subinterval and averages their function values.
Before calculating, identify the partition, find its width, and mark exactly which table entries the requested method needs. Do not include a value just because it appears in the table.
Reading the Table Before Calculating
Suppose the partition points are \(x_0,x_1,\ldots,x_n\), listed from left to right. For an equal-width partition, \(x_i=a+i\Delta x\). The table must include the sample values required by the chosen method:
- Left sum: use \(f(x_0), f(x_1), \ldots, f(x_{n-1})\). The rightmost endpoint \(x_n\) is not a left endpoint of any subinterval.
- Right sum: use \(f(x_1), f(x_2), \ldots, f(x_n)\). The leftmost endpoint \(x_0\) is not used.
- Midpoint sum: use the value at the center of each subinterval. Endpoint values alone are not enough to calculate this sum.
- Trapezoidal sum: use both endpoint values on each subinterval. An interior partition value is used for the two adjacent trapezoids.
For a common width \(\Delta x\), the formulas are:
The trapezoidal sum can be written by adding one trapezoid at a time, or by using the equivalent weighted formula:
These are numerical approximations, not automatically exact values of the integral. A sum method uses only the listed samples and the chosen geometric shapes; the function may behave differently between sample points.
Worked Example: Calculate All Four Methods
Worked Example: Calculate All Four Methods
The table gives values of \(f\) at the partition points and at the interval midpoints. Use the three subintervals from \(x=0\) to \(x=6\) to calculate \(L_3\), \(R_3\), \(M_3\), and \(T_3\).
| \(x\) | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|---|
| \(f(x)\) | 2 | 3 | 4 | 5 | 3 | 2 | 1 |
There are three subintervals, so their common width is:
The subintervals are \([0,2]\), \([2,4]\), and \([4,6]\). For the left sum, select the left endpoints \(0,2,4\):
For the right sum, select the right endpoints \(2,4,6\):
The midpoint of each subinterval is \(1\), \(3\), or \(5\). The table provides all three midpoint values, so:
For the trapezoidal sum, average the endpoint heights on each subinterval, then multiply each average by the width:
Check: with equal widths, the trapezoidal sum is the average of the left and right sums. Here, \((L_3+R_3)/2=(18+16)/2=17\), which agrees with the trapezoidal calculation.
Why the Trapezoidal Weights Work
The weighted trapezoid formula follows by adding the trapezoid areas on consecutive subintervals. On \([x_{i-1},x_i]\), the trapezoid has parallel side heights \(f(x_{i-1})\) and \(f(x_i)\), and width \(\Delta x\). Its signed area is the width times the average of those heights.
Adding this expression for all \(n\) subintervals gives the trapezoidal approximation. The first table value, \(f(x_0)\), appears only in the first trapezoid; the last, \(f(x_n)\), appears only in the last. Each interior value appears twice, because it is an endpoint of each of two neighboring trapezoids. Therefore:
This derivation explains the endpoint weights: the interior entries receive a coefficient of \(2\) because they occur in two trapezoids. The formula calculates the sum of those trapezoid areas; it does not claim that the function itself is linear on each subinterval.
Worked Example: Use Only the Midpoint Entries
Worked Example: Use Only the Midpoint Entries
A table lists values at the boundaries \(x=2,4,6,8\) and at the midpoints \(x=3,5,7\). Calculate the midpoint approximation on \([2,8]\).
| \(x\) | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|---|---|
| \(f(x)\) | 6 | 4 | 2 | 1 | 3 | 5 | 7 |
The partition has three equal subintervals, each of width:
The intervals are \([2,4]\), \([4,6]\), and \([6,8]\), whose midpoints are \(3,5,7\). Use the values at those points, not the values at the boundaries:
Check: the selected points \(3,5,7\) are each exactly halfway between the endpoints of their corresponding intervals: \((2+4)/2=3\), \((4+6)/2=5\), and \((6+8)/2=7\). The table values at those points sum to \(10\), and multiplying by the common width \(2\) gives \(20\).
Worked Example: A Trapezoidal Sum from Endpoint Data
Worked Example: A Trapezoidal Sum from Endpoint Data
Use the table to calculate the trapezoidal approximation on the four equal subintervals from \(x=1\) to \(x=9\).
| \(x\) | 1 | 3 | 5 | 7 | 9 |
|---|---|---|---|---|---|
| \(f(x)\) | 2 | 6 | 4 | 8 | 10 |
First find the common width:
Use the average of the two endpoint heights on each interval:
Check: using the weighted formula gives the same result:
The two calculations agree. The endpoint entries \(2\) and \(10\) appear once in the weighted formula, while each of the three interior entries appears twice.
A Common Table-Reading Mistake
A frequent error is to use every listed endpoint value in a left sum. The last endpoint is in the table, but it is not the left endpoint of any subinterval. For example, suppose the partition is \(0,2,4\), with \(f(0)=1\), \(f(2)=3\), and \(f(4)=5\). There are two subintervals, each of width \(2\).
The incorrect calculation includes all three entries:
But the two left endpoints are \(0\) and \(2\), so the correct left sum is:
Correction: write down the subintervals first. For \([0,2]\) and \([2,4]\), the left endpoints are \(0\) and \(2\); \(4\) is the right endpoint of the final interval and must not be added to the left sum.
Practical Checks Before You Submit
Table-based calculations are easier to verify when you keep the selection and the arithmetic separate. First identify the method and its sample points. Then copy only those function values, calculate the width, and perform the multiplication. Keep the selected values visible in your work so that an indexing error can be spotted.
- Confirm that the number of subintervals matches the partition.
- For a midpoint sum, check that every selected \(x\)-value is the center of its interval.
- For a left or right sum, do not confuse the endpoints of adjacent intervals.
- For a trapezoidal sum, ensure that each interval uses both of its endpoint values.
- If the table does not contain a method’s required sample values, the table alone does not supply the information needed for that calculation.
When a problem assigns units to \(x\) and \(f(x)\), the approximation has units formed by multiplying those units. For instance, if \(x\) is measured in minutes and \(f(x)\) in meters per minute, then \(\Delta x\,f(x)\) is measured in meters. Include units in the final answer when the problem provides them.
Check Your Understanding
For each question, identify the sample values to use before calculating.
- A table gives \(f(x)\) at \(x=0,2,4,6\). On this partition, which entries are used for the left sum, and which are used for the right sum?
- A partition has three equal subintervals from \(x=1\) to \(x=7\). What is the width of each subinterval?
- For a midpoint sum on \([0,6]\) divided into three equal intervals, which three \(x\)-values must the table provide?
- In a trapezoidal sum with four equal subintervals, why does each interior function value receive twice the weight of either endpoint in the weighted formula?
- For \(x=0,1,2\) with \(f(0)=2\), \(f(1)=4\), and \(f(2)=6\), calculate the trapezoidal sum and check it using the weighted formula.
